In a binomial distribution , the probability of getting asuccess is `1/4` and the standard deviation is 3 . Then its mean is

Options

A. 6
B. 8
C. 12
D. 48

Answer

To solve the problem step by step, we will use the properties of a binomial distribution.

### Step 1: Identify the given values
We are given:
- Probability of success, \( p = \frac{1}{4} \)
- Standard deviation, \( \sigma = 3 \)

### Step 2: Recall the formulas
In a binomial distribution:
- The variance \( \sigma^2 \) is given by the formula:
\[ \sigma^2 = n \cdot p \cdot q \]
where \( q = 1 - p \).

- The mean \( \mu \) is given by:
\[ \mu = n \cdot p \]

### Step 3: Calculate \( q \)
Since \( p = \frac{1}{4} \):
\[ q = 1 - p = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Substitute the values into the variance formula
We know that the standard deviation \( \sigma = 3 \), so:
\[ \sigma^2 = 3^2 = 9 \]
Now, substituting into the variance formula:
\[ 9 = n \cdot p \cdot q \]
Substituting \( p \) and \( q \):
\[ 9 = n \cdot \frac{1}{4} \cdot \frac{3}{4} \]

### Step 5: Simplify the equation
This simplifies to:
\[ 9 = n \cdot \frac{3}{16} \]
To isolate \( n \), multiply both sides by \( \frac{16}{3} \):
\[ n = 9 \cdot \frac{16}{3} = 48 \]

### Step 6: Calculate the mean
Now that we have \( n \), we can calculate the mean:
\[ \mu = n \cdot p = 48 \cdot \frac{1}{4} = 12 \]

### Conclusion
The mean of the binomial distribution is \( 12 \).